ISSN:

1534-0392

eISSN:

1553-5258

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## Communications on Pure & Applied Analysis

May 2017 , Volume 16 , Issue 3

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*+*[Abstract](1939)

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**Abstract:**

For the 2-D quasilinear wave equation $\sum\nolimits_{i,j = 0}^2 {{g_{ij}}} (\nabla u)\partial _{ij}^2u = 0$, whose coefficients are independent of the solution $u$, the blowup result of small data solution has been established in ^{[1,2]} when the null condition does not hold as well as a generic nondegenerate condition of initial data is assumed. In this paper, we are concerned with the more general 2-D quasilinear wave equation $\sum\nolimits_{i,j = 0}^2 {{g_{ij}}} (u,\nabla u)\partial _{ij}^2u = 0$, whose coefficients depend on $u$ and $\nabla u$ simultaneously. When the first weak null condition is not fulfilled and a suitable nondegenerate condition of initial data is assumed, we shall show that the small data smooth solution $u$ blows up in finite time, moreover, an explicit expression of lifespan and blowup mechanism are also established.

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**Abstract:**

In this paper, we investigative the large time decay and stability to any given global smooth solutions of the 3D incompressible inhomogeneous MHD systems. We prove that given a solution $(a, u, B)$ of (2), the velocity field and the magnetic field decay to zero with an explicit rate, for $u$ which coincide with incompressible inhomogeneous Navier-Stokes equations ^{[1]}. In particular, we give the decay rate of higher order derivatives of $u$ and $B$ which are useful to prove our main stability result. For a large solution of (2) denoted by $(a, u, B)$, we show that a small perturbation of the initial data still generates a unique global smooth solution and the smooth solution keeps close to the reference solution $(a, u, B)$. At last, we should mention that the main results in this paper are concerned with large solutions.

*+*[Abstract](1778)

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**Abstract:**

This paper is concerned with a nonlocal dispersal susceptible-infected-susceptible (SIS) epidemic model with Dirichlet boundary condition, where the rates of disease transmission and recovery are assumed to be spatially heterogeneous. We introduce a basic reproduction number $R_0$ and establish threshold-type results on the global dynamic in terms of *R*_{0}. More specifically, we show that if the basic reproduction number is less than one, then the disease will be extinct, and if the basic reproduction number is larger than one, then the disease will persist. Particularly, our results imply that the nonlocal dispersal of the infected individuals may suppress the spread of the disease even though in a high-risk domain.

*+*[Abstract](1893)

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**Abstract:**

We consider the Neumann and Dirichlet problems for second-order linear elliptic equations

in a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$, $n \geq 2$, where $A: \mathbb{R}^n \to \mathbb{R}^{n^2}$, $b: \Omega \to \mathbb{R}^n$ and $\lambda \geq 0$ are given. Some $W^{1, 2}$-estimates have been already known, provided that $A \in L^\infty(\Omega)^{n^2}$ and $b \in L^r(\Omega)^n$, where $n \leq r < \infty$ if $n \geq 3$ and $2 < r < \infty$ if $n=2$. Under more regularity assumptions on $A$ and $\Omega$, we establish the existence and uniqueness of weak solutions satisfying $W^{1, p}$-estimates. Our $W^{1, p}$-estimates are uniform on $\lambda \geq 0$ for the case of the Dirichlet problems. For the Neumann problems, the $W^{1, p}$-estimates are uniform with respect to $\lambda \geq 0$ if $f$ and $g$ satisfy some compatibility conditions. These uniform estimates allow us to obtain strong stability results in $W^{1, p}$ with respect to $\lambda $ for the Neumann and Dirichlet problems.

*+*[Abstract](1899)

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**Abstract:**

In this paper, we consider the following perturbed nonlocal elliptic equation

where $\Omega$ is a smooth bounded domain in $\mathbb{R}{^N}$, $\lambda$ is a real parameter and $g$ is a non-odd perturbation term. If $f$ is odd in $u$ and satisfies various superlinear growth conditions at infinity in $u$, infinitely many solutions are obtained in spite of the lack of the symmetry of this problem for any $\lambda\in \mathbb{R}$. The results obtained in this paper may be seen as natural extensions of some classical theorems to the case of nonlocal operators. Moreover, the methods used in this paper can be also applied to obtain some new results for the classical Laplace equation with Dirichlet boundary conditions.

*+*[Abstract](1393)

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**Abstract:**

We obtain sufficient conditions for nonexistence of positive solutions to some nonlinear parabolic inequalities with coefficients possessing singularities on unbounded sets.

*+*[Abstract](2200)

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**Abstract:**

Novel global weighted parabolic Sobolev estimates, weighted mixed-norm estimates and a.e. convergence results of singular integrals for evolution equations are obtained. Our results include the classical heat equation

the harmonic oscillator evolution equation

and their corresponding Cauchy problems. We also show weighted mixed-norm estimates for solutions to degenerate parabolic extension problems arising in connection with the fractional space-time nonlocal equations $(\partial_t-\Delta)^su=f$ and $(\partial_t-\Delta+|x|^2)^su=f$, for $0 < s < 1$.

*+*[Abstract](1917)

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**Abstract:**

In this paper, we establish the $L^p$ estimates for the maximal functions associated with the multilinear pseudo-differential operators. Our main result is Theorem 1.2. There are several major different ingredients and extra difficulties in our proof from those in Grafakos, Honzík and Seeger ^{[15]} and Honzík ^{[22]} for maximal functions generated by multipliers. First, in order to eliminate the variable $x$ in the symbols, we adapt a non-smooth modification of the smooth localization method developed by Muscalu in ^{[26,30]}. Then, by applying the inhomogeneous Littlewood-Paley dyadic decomposition and a discretization procedure, we can reduce the proof of Theorem 1.2 into proving the localized estimates for localized maximal functions generated by discrete paraproducts. The non-smooth cut-off functions in the localization procedure will be essential in establishing localized estimates. Finally, by proving a key localized square function estimate (Lemma 4.3) and applying the good-$\lambda$ inequality, we can derive the desired localized estimates.

*+*[Abstract](1723)

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**Abstract:**

We prove weighted Lorentz estimates of the Hessian of strong solution for nondivergence linear elliptic equations $a_{ij}(x)D_{ij}u(x)=f(x)$. The leading coefficients are assumed to be measurable with respect to one variable and have small BMO semi-norms with respect to the other variables. Here, an approximation method, Lorentz boundedness of the Hardy-Littlewood maximal operators and an equivalent representation of Lorentz norm are employed.

*+*[Abstract](2505)

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**Abstract:**

We study a two player zero-sum tug-of-war game with varying probabilities that depend on the game location x. In particular, we show that the value of the game is locally asymptotically Hölder continuous. The main difficulty is the loss of translation invariance. We also show the existence and uniqueness of values of the game. As an application, we prove that the value function of the game converges to a viscosity solution of the normalized *p*(*x*) -Laplacian.

*+*[Abstract](1574)

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**Abstract:**

In this paper, we study a fully nonlinear inverse curvature flow in Euclidean space, and prove a non-collapsing property for this flow using maximum principle. Precisely, we show that upon some conditions on speed function, the curvature of the largest touching interior ball is bounded by a multiple of the speed.

*+*[Abstract](1782)

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**Abstract:**

Motivated by relevant physical applications, we study Schrödinger equations with state-dependent potentials. Existence, localization and multiplicity results are established for positive standing wave solutions in the case of oscillating potentials. To this aim, a localized Pucci-Serrin type critical point theorem is first obtained. Two examples are then given to illustrate the new theory.

*+*[Abstract](1952)

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**Abstract:**

Improved Trudinger-Moser-Adams type inequalities in the spirit of Lions were recently studied in ^{[21]}. The main purpose of this paper is to prove the equivalence of these versions of the Trudinger-Moser-Adams type inequalities and to set up the relations of these Trudinger-Moser-Adams best constants. Moreover, using these identities, we will investigate the existence and nonexistence of the optimizers for some Trudinger-Moser-Adams type ineq

*+*[Abstract](1960)

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**Abstract:**

In this paper, we study the following quadratically coupled Schrödinger system:

where $\Omega\subset\mathbb{R}^6$ is a smooth bounded domain, $-\lambda (\Omega) < \lambda_1, \lambda_2 < 0, \mu_1, \mu_2, \alpha, \gamma>0$, and $\lambda (\Omega)$ is the first eigenvalue of $-\Delta$ with the Dirichlet boundary condition. The main difficulty to investigate this kind of equations is caused by the fact that all the quadratic nonlinearities, including the coupling terms, are of critical growth. By the methods used in [Zhenyu Guo, Positive ground state solutions of a nonlinearly coupled Schrödinger system with critical exponents in [Zhenyu Guo, Positive ground state solutions of a nonlinearly coupled Schrödinger system with critical exponents in $\mathbb{R}^4$, *J. Math. Anal. Appl.*, 430(2):950-970, 2015], the existence of positive ground state solutions of the system is established with more ingenious hypotheses.

*+*[Abstract](3193)

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**Abstract:**

This article is devoted to the discussion of the boundary layer which arises from the one-dimensional parabolic elliptic type Keller-Segel system to the corresponding aggregation system on the half space case. The characteristic boundary layer is shown to be stable for small diffusion coefficients by using asymptotic analysis and detailed error estimates between the Keller-Segel solution and the approximate solution. In the end, numerical simulations for this boundary layer problem are provided.

*+*[Abstract](2140)

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**Abstract:**

An attraction-repulsion chemotaxis model with nonlinear chemotactic sensitivity functions and growth source is considered. The global-in-time existence and boundedness of solutions are proved under some conditions on the nonlinear sensitivity functions and growth source function. Our results improve the earlier ones for the linear sensitivity functions.

*+*[Abstract](1942)

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**Abstract:**

In this paper, we consider the critical nonlinear Schrödinger equations in ${\mathbb{R}^2}$ with an oscillating nonlinearity, in a radial geometry. We numerically investigate the influence of the oscillations on the time of existence for the corresponding solution, on the spirit of the recent result of Cazenave and Scialom. It can be observed that the solution converges to the solution of a limit equation obtained with the weak limit of the oscillatory term, starting either with Gaussian data as well as standing waves solutions.

*+*[Abstract](1886)

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**Abstract:**

We study the second order nonlinear differential equation

where $\alpha_{i}, \beta_{j}>0$, $a_{i}(x), b_{j}(x)$ are non-negative Lebesgue integrable functions defined in $\mathopen{[}0, L\mathclose{]}$, and the nonlinearities $g_{i}(s), k_{j}(s)$ are continuous, positive and satisfy suitable growth conditions, as to cover the classical superlinear equation $u"+a(x)u.{p} = 0$, with $p>1$.When the positive parameters $\beta_{j}$ are sufficiently large, we prove the existence of at least $2.{m}-1$positive solutions for the Sturm-Liouville boundary value problems associated with the equation.The proof is based on the Leray-Schauder topological degree for locally compact operators on open and possibly unbounded sets.Finally, we deal with radially symmetric positive solutions for the Dirichlet problems associated with elliptic PDEs.

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